# Square wave amplitude and harmonics

Thread Starter

#### chris32432

Joined Jan 26, 2020
3
Hi all,

I have a square wave, 2 V amplitude, 500 microsecond period. I found its fundamental frequency as 2000Hz (I think!). Now, since a square wave with unit amplitude of 1 V has the amplitude of its first harmonic as 4/pi V, can i assume that a 2 V amplitude wave has 8/pi as the amplitude of its first harmonic? The amplitude of the 3rd harmonic is therefore (1/3)(8/pi) V? Hence the frequency of the 3rd non-zero harmonic will be 5(fundamental frequency)=10 kHz?

Thank you for your attention

#### Audioguru again

Joined Oct 21, 2019
3,201
There is no such thing as a 1st harmonic. A square wave has no even numbered harmonics because its high and low times are the same. It has a strong 3rd harmonic and its higher odd-numbered harmonics have reduced amplitudes like this:

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Thread Starter

#### chris32432

Joined Jan 26, 2020
3
There is no such thing as a 1st harmonic. A square wave has no even numbered harmonics because its high and low times are the same. It has a strong 3rd harmonic and its higher odd-numbered harmonics have reduced amplitudes like this:
Thank you for your reply. It seems to have given more questions though If there is no such thing as a first harmonic, do we call it (i.e. the leftmost spectral line) the 'principal harmonic'?

#### Alec_t

Joined Sep 17, 2013
12,074
It's called the 'fundamental'.

#### Audioguru again

Joined Oct 21, 2019
3,201
The "fundamental" is the main frequency #1 on the graph. A sawtooth waveform has even and odd harmonics but a square wave has only odd-numbered harmonics.

#### joeyd999

Joined Jun 6, 2011
4,477
There is no such thing as a 1st harmonic.
Of course there is: it's the fundamental. What does it mean that a square wave has only odd harmonics if you don't include the first? It only has 3+?

And, just for completeness, the 2nd harmonic is considered the 1st overtone.

#### MisterBill2

Joined Jan 23, 2018
8,787
There is no such thing as a 1st harmonic. A square wave has no even numbered harmonics because its high and low times are the same. It has a strong 3rd harmonic and its higher odd-numbered harmonics have reduced amplitudes like this:
Certainly there is such a thing as a first harmonic, but most folks call it the fundamental. All harmonics are identified as n x the fundamental frequency, where "n" is an integer. It is an uncommon use of the term but certainly there is a first harmonic. We learned that in the second semester circuit analysis class, a LOOOONG time ago.

#### Audioguru again

Joined Oct 21, 2019
3,201
It is silly to say that 1kHz is a harmonic of 1kHz.
If it is a pure sinewave then its fundamental is 1kHz and it has no harmonics.
If the fundamental frequency is 1kHz then if it is not symmetrical there even harmonics of 2kHz, 4kHz, 6kHz and etc.
If it is a squashed and symmetrical sinewave then it has odd-numbered harmonics of 3kHz, 5kHz, 7kHz and etc.

#### joeyd999

Joined Jun 6, 2011
4,477
It is silly to say that 1kHz is a harmonic of 1kHz.
If it is a pure sinewave then its fundamental is 1kHz and it has no harmonics.
If the fundamental frequency is 1kHz then if it is not symmetrical there even harmonics of 2kHz, 4kHz, 6kHz and etc.
If it is a squashed and symmetrical sinewave then it has odd-numbered harmonics of 3kHz, 5kHz, 7kHz and etc.
A one-legged man can be considered to have legs. Just one of them.

#### MisterBill2

Joined Jan 23, 2018
8,787
It is silly to say that 1kHz is a harmonic of 1kHz.
If it is a pure sinewave then its fundamental is 1kHz and it has no harmonics.
If the fundamental frequency is 1kHz then if it is not symmetrical there even harmonics of 2kHz, 4kHz, 6kHz and etc.
If it is a squashed and symmetrical sinewave then it has odd-numbered harmonics of 3kHz, 5kHz, 7kHz and etc.
In post #1 the TS is very specific in talking about a SQUARE WAVE, which is not s sine wave. They are quite a bit different.

#### WBahn

Joined Mar 31, 2012
26,398
Thank you for your reply. It seems to have given more questions though If there is no such thing as a first harmonic, do we call it (i.e. the leftmost spectral line) the 'principal harmonic'?
It's an arbitrary semantic distinction upon which there is no universal agreement.

While the "first harmonic" is better known as the "fundamental frequency", the term "first harmonic" is perfectly well defined; it is just more widely used in some fields than others. The nth harmonic is at a frequency that is n times the fundamental frequency, thus the first harmonic is the component that is at the fundamental frequency. Part of the issue is that many people assume that "harmonic" can only mean something at a higher frequency than the fundamental, and thus tend to think that talking about a harmonic that is at the fundamental frequency makes no sense. These folks tend to use "fundamental" for the first harmonic and "harmonics" to mean only the higher harmonics, whereas people that use the term "first harmonic" for the fundamental tend to distinguish the other harmonics by calling them the "higher harmonics".

Both views have practical issues. In some fields there is high value in talking about the first fundamental separate from the higher harmonics, so it makes quite reasonable sense to use the very different words "fundamental" vs "harmonics" to distinguish them, whereas the other camp has to use the longer phrases "first harmonic" and "higher harmonics" to do so and if they leave out the qualifier then their meaning can get miscommunicated. But from a purely mathematical perspective there really isn't much point in making that distinction and making an artificial one leads to inconsistencies (that the people in the first camp get so used to overlooking that they don't even realize they are there). A case in point is the very plot in Post #2. The plot talks about harmonics and has a horizontal axis that identifies them numerically, including one for harmonic #1. Well, if there's no such thing as the first harmonic, then that plot needs to be corrected to reflect that.

• andrewmm
Thread Starter

#### chris32432

Joined Jan 26, 2020
3
Thank you all for your insightful replies! May I reiterate one of the original questions: If the first (fundamental) harmonic of a square wave with unit amplitude of 1 V has magnitude 4/pi V, does a 2V square wave have a fundamental harmonic of amplitude 8/pi V?

#### dendad

Joined Feb 20, 2016
3,835
This sounds like a question from a homework assignment.

#### MisterBill2

Joined Jan 23, 2018
8,787
Thank you all for your insightful replies! May I reiterate one of the original questions: If the first (fundamental) harmonic of a square wave with unit amplitude of 1 V has magnitude 4/pi V, does a 2V square wave have a fundamental harmonic of amplitude 8/pi V?
As the amplitude of a fundamental wave increases the amplitude of all of it's components increases by the same proportion. Any other change would result in a distortion of the actual waveform. Distortion is usually caused by some non-linear element. A LINEAR change does not produce any distortion, thus all of the components increase by the exact same proportion when there is no distortion.

#### WBahn

Joined Mar 31, 2012
26,398
Thank you all for your insightful replies! May I reiterate one of the original questions: If the first (fundamental) harmonic of a square wave with unit amplitude of 1 V has magnitude 4/pi V, does a 2V square wave have a fundamental harmonic of amplitude 8/pi V?
Go look at the mathematical definition for how the Fourier components are computed. What happens to all of them if the amplitude of the signal is changed by a factor k?

#### MisterBill2

Joined Jan 23, 2018
8,787
As the amplitude of a fundamental wave increases the amplitude of all of it's components increases by the same proportion. Any other change would result in a distortion of the actual waveform. Distortion is usually caused by some non-linear element. A LINEAR change does not produce any distortion, thus all of the components increase by the exact same proportion when there is no distortion.
OK, so much for the long explanation. Short answer id YES. . Longer answer: YES BUT it does not show, because the square wave is a sum and some of the terms are negative, so they cancel out the peaks, which is why the square wave is flat on top. If you use the math and generate the series that is the basis of a square wave it will become obvious.

#### andrewmm

Joined Feb 25, 2011
1,474
Just to clarify
harmonics are integer multiples of the fundamental .
some might have zero amplitude, but they are always there.

#### crutschow

Joined Mar 14, 2008
27,229
some might have zero amplitude, but they are always there.
Another semantic issue.
If the value of a harmonic is zero then, to paraphrase Gertrude, there's no there, there. #### MisterBill2

Joined Jan 23, 2018
8,787
Another semantic issue.
If the value of a harmonic is zero then, to paraphrase Gertrude, there's no there, there. Ok, while I did not say that part about them always "being there", they are always defined, which is quite different. And as mentioned, they may be inverted, that is, a negative quantity. The entire derivation of a square wave is appropriate for a third term calculus class on a cold rainy fall day.